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1 Adiabatic and Diabatic Representations Advanced kinetics Solution 6 April 8, 2016 1 Adiabatic and diabatic representations 1.1 Born-Oppenheimer approximation The time-independent Schr¨odingerequation for both electronic and nuclear degrees of freedom is H^ Ψ(r; R) = E Ψ(r; R); (1) where the full molecular Hamiltonian reads (in atomic units, whereh ¯ = e = me = 1=(4π0) = 1) H^ = T^e + V^e + T^N + V^N + V^eN (2) X 1 X 1 X 1 X Zi Zj X Zj = − r2 + − r2 + − : (3) 2 e;i jr − r j 2 M N;i jR − R j jr − R j i j>i i j i i j>i i j ij i j We use fr; reg to refer to electron coordinates and momenta and fR; rNg for nuclear co- ordinates and momenta. Zi are the nuclear charges and Mi are the masses of the nuclei i. (both being expressed as multiples of the electron charge and electron mass, respectively). Equation (2) provides a shorthand notation for the five terms in Eq. (3), namely electron ki- netic energy, electron-electron repulsion, nuclear kinetic energy, nuclear-nuclear repulsion and electron-nuclear attraction. By defining the electronic Hamiltonian H^e = T^e + V^e + V^eN; (4) we get the eigen-energies Eel(R) from the electronic Schr¨odingerequation: el H^e Ψ(r; R) = E (R) Ψ(r; R): (5) We now insert the wavefunction described below Ψ(r; R) = (r; R) χ(R); (6) into Eq. (1), we get el H^ (r; R) (r; R) χ(R) = (T^N + E (R) + V^N) (r; R) χ(R) (7) = E (r; R)χ(R): (8) The Laplacian in the nuclear kinetic energy operator of the full wavefunction can be rewritten as: 2 rN;iΨ(r; R) = rN;i [rN;i (r; R) χ(R)] = rN;i [χ(R) rN;i (r; R) + (r; R) rN;i χ(R)] 2 = rN;i χ(R) · rN;i (r; R) + χ(R) rN;i (r; R) 2 +rN;i (r; R) · rN;i χ(R) + (r; R) rN;i χ(R) 2 = (r; R) rN;i χ(R) + 2 rN;i (r; R) · rN;i χ(R) 2 +χ(R) rN;i (r; R): (9) 1 Advanced kinetics Solution 6 April 8, 2016 As a result, Eq. (7) can be written as: el H^ (r; R) (r; R) χ(R) = (E (R) + V^N) (r; R) χ(R) X 1 − [ (r; R) r2 χ(R) 2 M N;i i i 2 +2 rN;i (r; R) · rN;iχ(R) + rN;i (r; R) χ(R)]: (10) These last three terms in Eq. (10) arise from the nuclear kinetic energy operator acting on the full electronic-nuclear wavefunction. While the first of these three terms represents the nuclear kinetic energy, the other two terms contain derivatives of the electronic wave function with respect to the nuclear coordinates. As these derivatives carry the inverse masses of the nuclei as prefactors their relative weight is usually quite small and we can neglect them. If we neglect these terms (which is known as the Born-Oppenheimer approximation), we end up with a time-independent Schr¨odingerequation for the nuclei: (note that the T^N below is the operator acting only on the nuclear coordinates) el H^ (r; R) χ(R) = (T^N + E (R) + V^N(R)) χ(R) = E χ(R): (11) However, the derivative terms can become large when potential energy surfaces cross. In such regions of the potential, small changes in the nuclear geometry can cause significant changes of the character of the electronic wave function. In a time-domain perspective, this means that the electronic dynamics evolve on a time scale comparable to the nuclear dynamics as the energies of the electronic states approach each other in the crossing region of potential energy surfaces. Thus the electronic level spacings become comparable to typical vibrational energy-level spacings. 1.2 Adiabatic representation We now substitute 1 X Ψ(r; R) = n(r; R) χn(R); (12) n=0 in the time-independent Schr¨odingerequation and project onto h m(r; R)j: X X h m(r; R)j H^ (r; R) j n(r; R)i jχn(R)i = h m(r; R)j H^ (r; R) j n(r; R)i jχn(R)i n n | {z } X = Hmn(R) jχn(R)i ; (13) n within which X X ^ el ^ Hmn(R) jχn(R)i = [TN + En (R) + VN(R)]δmn n n X 1 (1) (2) + (2 T (R) · r + T ) jχ (R)i ; 2 M mn;i N;i mn;i n i i = E jχm(R)i with the identities: (14) (2) 2 (1) Tmn;i ≡ h mjrN;i ni ;Tmn;i ≡ h mjrN;i ni : (15) 2 Advanced kinetics Solution 6 April 8, 2016 Here the braket notation refers to integration over the electronic coordinates only. The differen- (1) tiation works similar as in Eq. (9). We now simplify the vectors Tmn by evaluating r h mj ni and taking the electronic wave functions to be orthonormal at all nuclear geometries, i.e. h mj ni = δmn. Thus rN;i h mj ni = hrN;i mj ni + h mjrN;i ni = 0; (16) hrN;i mj ni = − h mjrN;i ni : (17) If we choose real-valued electronic wavefunctions, it follows from the above equation that (1) Tnn;i = 0. If we now keep the diagonal elements of Eq. (14) only, we end up with an expression (2) similar to Eq. (11) that contains an additional term Tnn;i: ! X 1 (2) T^ + Eel(R) + V^ (R) + T χ (R) = E χ (R): (18) N n N 2 M nn;i n n i i (2) The derivative terms Tnn;i give an additional contribution to the energy. However, they do not give rise to a coupling between different electronic states, which is contained in the off-diagonal elements in Eq. (14). The difficulty to calculate these off-diagonal elements is the main reason to only employ the adiabatic representation whenever the off-diagonal elements are sufficiently small. 1.3 Diabatic representation We insert the following ansatz for the wavefunction 1 X (0) Ψ(r; R) = n(r; R0) χn (R): (19) n=0 in the time-independent Schr¨odingerequation (using Eq. 3 on the Exercise sheet): 1 1 1 X ^ ^ (0) X (0) ^ X (0) n(r; R0)(TN + VN)χn (R) + χn (R) He n(r; R0) = E n(r; R0) χn (R): (20) n=0 n=0 n=0 Note that the kinetic energy operator of the nuclei T^N is not acting on the electronic wave- functions n(r; R0) anymore, as those are defined for a fixed set of nuclear coordinates R0. We now project onto h m(r; R0)j: 1 X ^ ^ (0) (0) [(TN + VN)δmn + He;mn]χn (R) = E χm (R); (21) n=0 where He;mn = h m(r; R0)j H^e(r; R) j n(r; R0)i ; (22) with H^e being the electronic Hamiltonian defined in Eq. (4). As the electronic wavefunctions are defined for a fixed nuclear geometry R0 they are eigen- functions of the electronic Hamiltonian at a fixed nuclear geometry, i.e. ^ el He(r; R0) n(r; R0) = En (R0) n(r; R0): (23) 3 Advanced kinetics Solution 6 April 8, 2016 Therefore we rewrite the electronic Hamiltonian using V^ ≡ V^e + V^eN H^e(r; R) = T^e(r) + V^ (r; R) = H^e(r; R0) − V^ (r; R0) + V^ (r; R); (24) where we have used H^e(r; R0) − V^ (r; R0) = T^e(r) + V^e(r) + V^eN (r; R0) − [V^e(r) + V^eN (r; R0)] = T^e(r); (25) | {z } 0 and note that T^e(r; R0) = T^e(r); V^e(r; R0) = V^e(r). Now we can simplify the matrix elements He;mn: He;mn = h m(r; R0)j H^e(r; R0) − V^ (r; R0) + V^ (r; R) j n(r; R0)i el ^ ^ = En (R0) δmn + h m(r; R0)j V (r; R) − V (r; R0) j n(r; R0)i : (26) We use the definition Umn = h m(r; R0)j V^ (r; R) − V^ (r; R0) j n(r; R0)i ; (27) as given in the task and insert Eq. (26) in Eq. (21), which yields: X h ^ ^ el i (0) (0) fTN + VN + En (R0)gδnm + Umn χn = E χm ; (28) n ^ ^ ^ el or in the other notation (Eq. 12.18 in the book of Tannor), where Tnn = TN + VN + En (R0) X h ^ i (0) (0) Tnnδnm + Umn χn = E χm : (29) n This expression is often refered to as the time-independent Schr¨odingerequation in a diabatic basis. More specifically, our choice of the basis in Eq. (19) is often called a crude adiabatic basis. Compare Eq. (28) with the adiabatic form of the time-independent Schr¨odingerequa- (1) tion in Eq. (14). Both describe the same physics, but instead of derivative couplings (Tmn and (2) Tmn) between the potential energy surfaces of the different electronic states, we now deal with coordinate couplings Umn. The diagonal elements of Umn are referred to as diabatic potentials, where the off-diagonal elements introduce the couplings between the diabatic surfaces. The diagonalization of U then gives the adiabatic potential energy surfaces. The drawback of this diabatic basis is that the electronic wave function has to be calculated at many different ge- ometries R0. This results in a much larger number of basis functions required for an accurate calculation as compared to the adiabatic basis (Eq.
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